Learning to Count Isomorphisms with Graph Neural Networks

Subgraph Isomorphism is a fundamental problem in graph theory. It has many applications in social network analysis, molecular investigations, knowledge graphs, etc. Given a Query Graph and a Data Graph, the target of Subgraph Isomorphism, i.e., Subgraph Matching, is to determine if this Query Graph is isomorphic to any subgraph of the Data Graph. This work proposes a new type of Query Graph, combined with multiple general Query Graphs. We call it Compulsory-Optional Query Graph (CO Query Graph). This new type of Query Graph contains all the vertices in the combined general Query Graph, and each vertex corresponds to a search priority. Based on CO Query Graph, the previous multiple match processes can be reduced to one. It tremendously improves search efficiency. The Subgraph Isomorphism based on this new kind of Query Graph is an extension and improvement of the previous Subgraph Isomorphism studies. We propose a backtracking-pruning-based CO solver (BPC). This algorithm builds on the backtracking-pruning framework. BPC modifies the output criterion and matching conditions to satisfy the CO query context. A case study of real-world graph data illustrates that BPC built on CO Query Graph is more efficient than conventional Query Graphs. To verify the effectiveness of our method, we conducted experiments on the synthetic graph and real-world data. The results show that the BPC can significantly reduce the search space and improve the search efficiency in the recursive calls and the response time. Experiments resulting from synthetic graph data analysis allow us to primarily identify the critical factor that affects the efficiency of the BPC primarily.

Search-based graph queries, such as finding short paths and isomorphic subgraphs, are dominated by memory latency. If input graphs can be partitioned appropriately, large cluster-based computing platforms can run these queries. However, the lack of compute-bound processing at each vertex of the input graph and the constant need to retrieve neighbors implies low processor utilization. Furthermore, graph classes such as scale-free social networks lack the locality to make partitioning clearly effective. Massive multithreading is an alternative architectural paradigm, in which a large shared memory is combined with processors that have extra hardware to support many thread contexts. The processor speed is typically slower than normal, and there is no data cache. Rather than mitigating memory latency, multithreaded machines tolerate it. This paradigm is well aligned with the problem of graph search, as the high ratio of memory requests to computation can be tolerated via multithreading. In this paper, we introduce the multithreaded graph library (MTGL), generic graph query software for processing semantic graphs on multithreaded computers. This library currently runs on serial machines and the Cray MTA-2, but Sandia is developing a run-time system that will make it possible to run MTGL-based code on symmetric multiprocessors. We also introduce a multithreaded algorithm for connected components and a new heuristic for inexact subgraph isomorphism We explore the performance of these and other basic graph algorithms on large scale-free graphs. We conclude with a performance comparison between the Cray MTA-2 and Blue Gene/Light for s-t connectivity.

Graph Neural Networks (GNNs) have achieved remarkable success in analyzing graph-structured data, with their performance dependent on the graph structure. However, models trained on high-quality graph structures often suffer a significant performance drop when evaluated on perturbed graphs. Existing methods tackle this problem by improving the robustness of GNNs, but they often overlook representation deviation caused by structural changes. To address this limitation, we propose an attribute-guided dynamic prompt learning model that generates prompt vectors to approximate the intrinsic information of nodes. With these prompt vectors, the trained GNNs are expected to maintain their performance under perturbed graph structures. Unlike previous prompt-based methods that learn unified prompt vectors for all nodes, we obtain node-level prompts by encoding node attributes that provide unique information. Given the diversity of perturbed graph structures during inference, we introduce a structure-aware adaptation mechanism that adjusts the prompt vectors based on the input graph. Furthermore, we apply gradient-based attacks to generate perturbed graphs, encouraging the model to generalize to unseen structures. Extensive experiments on multiple benchmark datasets demonstrate the effectiveness and robustness of our model.

In general, graph neural networks (GNNs) adopt the message-passing scheme to capture the information of a node (i.e., nodal attributes, and local graph structure) by iteratively transforming, aggregating the features of its neighbors. Nonetheless, recent studies show that the performance of GNNs can be easily hampered by the existence of abnormal or malicious nodes due to the vulnerability of neighborhood aggregation. Thus it is necessary to learn anomaly-resistant GNNs without the prior knowledge of ground-truth anomalies, given the fact that labeling anomalies is costly and requires intensive domain knowledge. Though removing anomalies through unsupervised anomaly detection methods could be a possible solution, it may render unreasonable GNN model performance on target tasks due to the non-differentiable gap between the two learning procedures. In order to keep the effectiveness of GNNs on anomaly-contaminated graphs, in this paper, we propose a new framework named RARE-GNN (Reinforced Anomaly-REsistant Graph Neural Networks) which can detect anomalies from the input graph and learn anomaly-resistant GNNs simultaneously. Extensive experiments on real-world datasets demonstrate the effectiveness of the proposed framework.

Graph neural networks (GNNs) find applications in various domains such as computational biology, natural language processing, and computer security. Owing to their popularity, there is an increasing need to explain GNN predictions since GNNs are black-box machine learning models. One way to address this issue involves using counterfactual reasoning where the objective is to alter the GNN prediction by minimal changes in the input graph. Existing methods for counterfactual explanation of GNNs are limited to instance-specific local reasoning. This approach has two major limitations of not being able to offer global recourse policies and overloading human cognitive ability with too much information. In this work, we study the global explainability of GNNs through global counterfactual reasoning. Specifically, we want to find a small set of representative counterfactual graphs that explains all input graphs. Towards this goal, we propose GCF Explainer , a novel algorithm powered by vertex-reinforced random walks on an edit map of graphs with a greedy summary . Extensive experiments on real graph datasets show that the global explanation from GCF Explainer provides important high-level insights of the model behavior and achieves a 46.9% gain in recourse coverage, a 9.5% reduction in recourse cost compared to the state-of-the-art local counterfactual explainers. We also demonstrate that GCF Explainer generates explanations that are more consistent with input dataset characteristics, and is robust under adversarial attacks. In addition, K-GCF Explainer , which incorporates a graph clustering component into GCF Explainer , is introduced as a more competitive extension for datasets with a clustering structure, leading to superior performance in three out of four datasets in the experiments and better scalability.

Graph Neural Networks (GNNs) have demonstrated significant success in various graph-related tasks. However, their performance is highly dependent on the availability of sufficient, class-balanced annotated data and an accurate graph structure. GNNs are vulnerable to substantial performance degradation when confronted with limited annotated samples, class imbalance, or noisy graph structures. First, a Dual GNN learning framework is introduced for semi-supervised node classification under limited supervision. The framework comprises two GNN-based node prediction modules: a primary module, which uses the input graph structure to induce standard node embeddings and predictions, and an auxiliary module, which leverages spectral clustering to construct a new graph structure and learn new node embeddings and predictions. These modules collaborate to enable end-to-end learning of discriminative node representations. Next, a graph augmentation method called Graph Dual Mixup (GDM) is proposed for graph classification with scarce labels. GDM employs a graph structural auto-encoder to learn structural embeddings, applying mixup in the learned structural embedding space to generate new graph structures. Additionally, mixup is applied to input node features, generating node features for new graph instances. Together, these generated node features and graph structures contribute new graphs that increase the size and diversity of the labeled dataset, improving classification performance. To enhance robustness against adversarial attacks on graph structures, an Efficient Low-Rank GNN is introduced. This approach learns robust low-rank sparse graph structures through a two-stage process. In the first stage, Singular Value Decomposition is used to estimate a low-rank approximation of the graph structure. This estimate is refined in the second stage by jointly learning a low-rank sparse graph structure alongside the GNN model, resulting in enhanced robustness. Finally, to address class-imbalanced node classification, a Unified GNN Learning (Uni-GNN) framework is proposed. Uni-GNN integrates structural and semantic connectivity representations to extend the propagation of embeddings to non-adjacent structural neighbors and semantically similar nodes, facilitating the diffusion of discriminative information. Additionally, it incorporates a balanced pseudo-label generation mechanism to improve the representations of minority classes. Comprehensive evaluations on benchmark datasets demonstrate that these methods outperform state-of-the-art approaches and hold significant potential for advancing GNN learning with limited supervision.

Query graph construction aims to construct the correct executable SPARQL on the KG to answer natural language questions. Although recent methods have achieved good results using neural network-based query graph ranking, they suffer from three new challenges when handling more complex questions: 1) complicated SPARQL syntax, 2) huge search space, and 3) locally ambiguous query graphs. In this paper, we provide a new solution. As a preparation, we extend the query graph by treating each SPARQL clause as a subgraph consisting of vertices and edges and define a unified graph grammar called AQG to describe the structure of query graphs. Based on these concepts, we propose a novel end-to-end model that performs hierarchical autoregressive decoding to generate query graphs. The high-level decoding generates an AQG as a constraint to prune the search space and reduce the locally ambiguous query graph. The bottom-level decoding accomplishes the query graph construction by selecting appropriate instances from the preprepared candidates to fill the slots in the AQG. The experimental results show that our method greatly improves the SOTA performance on complex KGQA benchmarks. Equipped with pre-trained models, the performance of our method is further improved, achieving SOTA for all three datasets used.

Graph causal representation learning for out-of-distribution generalization.

Graph Neural Networks (GNNs) are the most common approach for learning complex relational data represented using graph data structures. Although GNNs are effective at learning representations of both nodes and graphs for a given task, the learning process is computationally expensive and as such, time and energy-inefficient. This paper investigates this challenge within the context of recent work on untrained graph representations that only train the solver model. We present Graph Vector Function Architecture (GVFA), a novel alternative to learning graph representations in GNNs that is based on hyperdimensional computing (HDC) principles. GVFA is a general zero-shot approach for graph and node representations without learning. As such, our representations are not task-specific and the computational costs of constructing them is substantially lower compared to learning-based GNN. Empirically, we demonstrate the expressiveness and generalization properties of different GVFA configurations. Our experimental results demonstrate that GVFA outperforms several classic GNNs on their benchmark datasets in terms of classification accuracy for both graph and node classification tasks, while also yielding a substantial reduction in training time.

The dramatic proliferation of sophisticated networks has resulted in a growing need for supporting effective querying and mining methods over such large-scale graph-structured data. At the core of many advanced network operations lies a common and critical graph query primitive: how to search graph structures efficiently within a large network? Unfortunately, the graph query is hard due to the NP-complete nature of subgraph isomorphism. It becomes even challenging when the network examined is large and diverse. In this paper, we present a high performance graph indexing mechanism, SPath, to address the graph query problem on large networks. SPath leverages decomposed shortest paths around vertex neighborhood as basic indexing units, which prove to be both effective in graph search space pruning and highly scalable in index construction and deployment. Via SPath, a graph query is processed and optimized beyond the traditional vertex-at-a-time fashion to a more efficient path-at-a-time way: the query is first decomposed to a set of shortest paths, among which a subset of candidates with good selectivity is picked by a query plan optimizer; Candidate paths are further joined together to help recover the query graph to finalize the graph query processing. We evaluate SPath with the state-of-the-art GraphQL on both real and synthetic data sets. Our experimental studies demonstrate the effectiveness and scalability of SPath, which proves to be a more practical and efficient indexing method in addressing graph queries on large networks.

Enhanced subgraph matching for large graphs using candidate region-based decomposition and ordering

Explainability is crucial for probing graph neural networks (GNNs), answering questions like "Why the GNN model makes a certain prediction?". Feature attribution is a prevalent technique of highlighting the explanatory subgraph in the input graph, which plausibly leads the GNN model to make its prediction. Various attribution methods have been proposed to exploit gradient-like or attention scores as the attributions of edges, then select the salient edges with top attribution scores as the explanation. However, most of these works make an untenable assumption - the selected edges are linearly independent - thus leaving the dependencies among edges largely unexplored, especially their coalition effect. We demonstrate unambiguous drawbacks of this assumption - making the explanatory subgraph unfaithful and verbose. To address this challenge, we propose a reinforcement learning agent, Reinforced Causal Explainer (RC-Explainer). It frames the explanation task as a sequential decision process - an explanatory subgraph is successively constructed by adding a salient edge to connect the previously selected subgraph. Technically, its policy network predicts the action of edge addition, and gets a reward that quantifies the action's causal effect on the prediction. Such reward accounts for the dependency of the newly-added edge and the previously-added edges, thus reflecting whether they collaborate together and form a coalition to pursue better explanations. It is trained via policy gradient to optimize the reward stream of edge sequences. As such, RC-Explainer is able to generate faithful and concise explanations, and has a better generalization power to unseen graphs. When explaining different GNNs on three graph classification datasets, RC-Explainer achieves better or comparable performance to state-of-the-art approaches w.r.t. two quantitative metrics: predictive accuracy, contrastivity, and safely passes sanity checks and visual inspections. Codes and datasets are available at https://github.com/xiangwang1223/reinforced_causal_explainer.

We investigate works under the propagation-based fake news detection domain, which recently seeks to improve performance through the use of Graph Neural Networks (GNNs). Generally, existing works argue that using GNNs can give results superior to what was obtained using classic graph-based methods. We agree with this argument given that GNNs are capable of gaining superior performance by leveraging node features. But we argue that existing works haven't identified the fact that the expressivity of GNNs is limited and bounded by node features. Existing works do not acknowledge that, by utilizing GNNs, they implicitly assume node features are strongly correlated to node labels. There are evidence that node features that have been employed do not necessarily correlate to node label. Instead of having a profound theoretical motivation, they have empirically observed that focusing on nodes features with strong feature-label correlation can increase predictive capability. This is a sub-optimal approach to view this problem, in fact, we argue that finding node features based on correlation is not practical or effective. Our first contribution is shifting readers from a node-level view i.e correlating node features with labels, to a graph-level view. In the graph-level view, we exploit the relationship between graph isomorphism and GNNs' expressivity which can be utilized to well understand and interpret the relation between node features and GNNs' expressivity. We conduct a wide range of experiments on basis of both node-level view and graph-level view and found graph-level view is more interpretable and strongly matches with results. Further, we gained insights on node features that wouldn't be obtainable by a node-level view. In order to have a fair and comprehensive analysis of node features, we built a unified dataset that includes a wide range of node features. Our results indicate, as we improve model accuracy on basis of the graph level view, models' generalizability decreases. We provide our hypothesis for this performance trade-off on the basis of the graph-level view. Our results and insights call for a much broader discussion on whether any sort of filtering method is effective. So, we conclude our work by providing readers with possible solutions that can be helpful to find harmony between node features and GNNs' expressivity.

Graph matching, as one of the most fundamental problems in graph database, has a wide range of applications. Due to the large scale of graph database and the hardness of graph matching, it is an attractive alternative to make use of the cloud to store massive data graphs and conduct the complex graph matching. To protect the privacy, data graphs are usually encrypted before being outsourced to the cloud. A few schemes have been proposed to support graph matching query over encrypted graphs. However, none of them can realize efficient subgraph extraction when the matched subgraph needs to be exactly located at the data graph. The graph user has to perform the complex subgraph isomorphism operation to extract the isomorphic subgraph from the matched data graph in state-of-the-art schemes. The time complexity of judging subgraph isomorphism is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(n!n^{2})$</tex-math></inline-formula> at most, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> is the number of vertices in the data graph. In order to solve this problem, we propose a privacy-preserving graph matching query scheme supporting quick subgraph extraction in this paper. In our design, two non-colluding cloud servers are adopted to accomplish the matching operation jointly. Neither of them can infer the plaintexts of graphs. The first cloud server can prune some impossible matched vertices from the data graph, and get a possible matched matrix to represent which vertices in the data graph might match with the vertices in the query graph. The second cloud server performs the related matrix operations based on its stored information and the information sent by the first cloud server. The two cloud servers jointly verify the correctness of the matched matrix. The graph user can directly and quickly extract the matrix of the subgraph isomorphic to the query graph from the data graph matrix based on the matched matrix, and further recover this subgraph. No subgraph isomorphism operation is involved in the procedure of subgraph extraction for the graph user. The time complexity of subgraph extraction is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(m^{2})$</tex-math></inline-formula> in our scheme, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> is the number of vertices in the query graph. The extensive experiments with real-world database demonstrate the efficiency of the proposed privacy-preserving graph matching scheme.

Subgraph matching problem is identifying a target subgraph in a graph. Graph neural network (GNN) is an artificial neural network model which is capable of processing general types of graph structured data. A graph may contain many subgraphs isomorphic to a given target graph. In this paper GNN is modeled to identify a subgraph that matches the target graph along with its characteristics. The simulation results show that GNN is capable of identifying a target sub-graph in a graph.